Consider a crystal under the perturbation of a weak electric field. We can write the total Hamiltonian as \(\hat{H}_{\mathrm {eff}}(\vec{k}(t)) = \hat{H}_{\mathrm {eff}}+\hat{H^{\prime }}\) , where \(\hat{H^{\prime }}\)  stands for the perturbation Hamiltonian. If the system is initially in the ground band, it will always stay in this band if the adiabatic condition is met. But now, we consider the case that there is still a small probability for particles to move from ground to excited band. So we can use the adiabatic perturbation theory with the perturbation Hamiltonian \(\hat{H^{\prime }} = -i\partial /\partial t\) . The corresponding ground band wave function up to the first order in the field strength satisfies [1]}
\(\begin{aligned}|u^\prime _-\rangle =|u_{-}({\vec{k}})\rangle - i \frac{\langle \hat{u}_{+}({\vec{k}}) | \partial _{t} u_{-}({\vec{k}})\rangle }{E_{-}-E_{+}}|u_{+}({\vec{k}})\rangle ,\end{aligned}\) 
